# On the Evolution of the Optimal Design of WDS: Shifting towards the Use of a Fractal Criterion

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background

## 3. Methodology

#### 3.1. OPUS

- (1)
- Uniform distribution: It is assumed that all pipes have the same flow, Therefore the total demand of each node is divided into the number of upstream pipes connected to it.
- (2)
- Proportional distribution: the flow of each pipe is proportional to H/L
^{2}, where H are the head losses in the pipe and L is its length. - (3)

#### 3.2. NSGA-II

- Minimum pressure at all demand nodes.
- Maximum pressure and maximum velocity for avoidance of operational inconveniences.
- Mass and energy conservation.
- Discrete pipe diameters.

#### 3.3. NSGA-II Methodology with OPUS Intermittent Feedback

#### 3.3.1. Calibration Process

#### 3.3.2. Preprocessing of OPUS Results

#### 3.3.3. NSGA-II Intermittent Retrofitting through OPUS

_{i}represents a pipe diameter of an individual obtained through NSGA-II optimization and ${f}_{3}^{-1}$(Q

_{i}) represents a pipe diameter of an individual obtained through OPUS with flow rates extracted from the EPANET hydraulic simulation with former individuals having diameters d

_{i}[5]. Finally, Paez et al. [5] have set a feedback frequency of m $\in $ (5, 50) for balancing a trade-off between PF quality and the algorithm’s convergence rate.

#### 3.4. Optimal WDSs Selection

- (1)
- A min(C), min(NRI) point corresponding to the simultaneously minimum cost (C) and minimum reliability (NRI) WDS arrangement.
- (2)
- A knee(C), knee(NRI) point corresponding to the knee cost and knee reliability (NRI) WDS arrangement.
- (3)
- A max(C), max(NRI) point corresponding to the maximum cost (C) and maximum reliability (NRI) WDS arrangement.

#### 3.5. Obtention of the Fractal Dimension

_{j}is the piezometric head value obtained by EPANET using instantaneous hydraulic simulation, ${d}_{ij}$ is the diameter of the pipe i incoming or outgoing from node j, ${Q}_{ij}$ is the flow rate through the aforementioned pipe i incoming or outgoing from node j, and ${D}_{j}$ is the demand at node j. The criteria will be refered to as the HGL, Diamater, and Flow Rate criterion, respectively.

^{2}coefficient for the linear regression should be close to 1 [33].

^{2}value greater than 0.99 is achieved given that a minimum number of iterations is made.

^{2}value before flattening.

#### 3.6. Water Distribution System Classification

## 4. Study Cases

#### 4.1. Hanoi

#### 4.2. Fossolo

_{min}= 40 m and the maximum velocity in each pipe is 1 m/s. Figure 4 shows the HGL for the chosen WDS configurations for Fossolo (Table S4, Figure S4 in Supplementary Materials).

#### 4.3. Balerma

_{s}= 0.0025 mm. The network includes a minimum pressure constraint of P

_{min}$=$ 20 m [37]. Figure 5 shows the HGL for the chosen WDS configurations for Balerma (Table S2, Figure S2 in Supplementary Materials).

#### 4.4. Modena

## 5. Results and Discussion

## 6. Conclusions and Future Work

^{3}), which is polynomial and desirable. The results do show tendencies, but these are of a very small order; in general, it is safe to suggest that the fractal dimensions of optimal designs for a given network are almost the same. Although this study only analyzed optimal networks, Jaramillo [31] also found fractal dimensions, although with a slightly different methodology, of non-optimal networks and did not observe the patterns here exemplified. That suggests that the limiting value could be unique to the optimal networks. Therefore, this paper has successfully developed a consistent methodology with a low computational cost that can be integrated to more costly algorithms as an optimality criterion that can reduce the overall running time.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Klingel, P.; Nestmann, F. From Intermittent to Continuous Water Distribution: A Proposed Conceptual Approach and a Case Study of Béni Abbès (Algeria). Urban Water J.
**2014**, 11, 240–251. [Google Scholar] [CrossRef] - Giudicianni, C.; Herrera, M.; di Nardo, A.; Carravetta, A.; Ramos, H.M.; Adeyeye, K. Zero-Net Energy Management for the Monitoring and Control of Dynamically-Partitioned Smart Water Systems. J. Clean. Prod.
**2020**, 252, 119745. [Google Scholar] [CrossRef] - Central Public Health and Environmental Engineering Organisation; Ministry of Urban Development. World Health Organisation Manual on Operation and Maintenance of Water Supply Systems; World Health Organization, UN: New Delhi, India, 2005. [Google Scholar]
- Saldarriaga, J.; Páez, D.; Cuero, P.; León, N. Optimal Power Use Surface for Design of Water Distribution Systems. In Proceedings of the WDSA 2012: 14th Water Distribution Systems Analysis Conference, Adelaide, Australia, 24 September 2012; p. 468. [Google Scholar]
- Páez, D.; Salcedo, C.; Garzón, A.; González, M.A.; Saldarriaga, J. Use of Energy-Based Domain Knowledge as Feedback to Evolutionary Algorithms for the Optimization of Water Distribution Networks. Water
**2020**, 12, 3101. [Google Scholar] [CrossRef] - Storn, R.; Price, K. Differential Evolution-A Simple and Efficient Heuristic for Global Optimization over Continuous Spaces. J. Glob. Optim.
**1997**, 11, 341–359. [Google Scholar] [CrossRef] - Savic, D.A.; Walters, G.A. Genetic Algorithms for Least-Cost Design of Water Distribution Networks. J. Water Resour. Plan. Manag.
**1997**, 123, 67–77. [Google Scholar] [CrossRef] - Rossman, L.A. EPANET 2 Users Manual; MicroImages: Cincinnati, OH, USA, 2000; Available online: https://epanet.es/wp-content/uploads/2012/10/EPANET_User_Guide.pdf (accessed on 8 November 2022).
- Reca, J.; Martínez, J.; López, R. A Hybrid Water Distribution Networks Design Optimization Method Based on a Search Space Reduction Approach and a Genetic Algorithm. Water
**2017**, 9, 845. [Google Scholar] [CrossRef][Green Version] - Saldarriaga, J.; Asce, A.M.; Páez, D.; Bohórquezboh´bohórquez, J.; Páez, N.; Juan, P.P.; Rincón, D.; Rincón, R.; Salcedo, C. Rehabilitation and Leakage Reduction on C-Town Using Hydraulic Criteria. J. Water Resour. Plann. Manag.
**2016**, 142, c4015013. [Google Scholar] [CrossRef] - Salcedo, C.A.; Saldarriaga, J. Use of Hydraulic Based-Criteria for the Reduction of the Solution Space Addressing the Problem of Optimal Valve Location within a WDS. In Proceedings of the XV Seminario Iberoamericano de Redes de Agua y Drenaje, SEREA2017, Bogotá, Colombia, 27–30 November 2017; pp. 1–9. Available online: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3113782 (accessed on 8 November 2022).
- Takahashi, S.; Saldarriaga, J.; Hernández, F.; Díaz, D.M.; Ochoa, S. An Energy Methodology for the Design of Water Distribution Systems. In Proceedings of the World Environmental and Water Resources Congress 2011: Bearing Knowledge for Sustainability, Palm Springs, CA, USA, 22 May 2011. [Google Scholar]
- Saldarriaga, J.; Páez, D.; Salcedo, C.; Cuero, P.; López, L.L.; León, N.; Celeita, D. A Direct Approach for the Near-Optimal Design of Water Distribution Networks Based on Power Use. Water
**2020**, 12, 1037. [Google Scholar] [CrossRef][Green Version] - Wu, I. Design of Drip Irrigation Main Lines. J. Irrig. Drain. Div.
**1975**, 101, 265–278. [Google Scholar] [CrossRef] - Featherstone, R.E.; El-Jumaily, K.K. Optimal Diameter Selection for Pipe Networks. J. Hydraul. Eng.
**1983**, 109, 221–234. [Google Scholar] [CrossRef] - Farmani, R.; Savic, D.A.; Walters, G.A. “EXNET” Benchmark Problem for Multi-Objective Optimization of Large Water Systems. In Proceedings of the Modelling and Control for Participatory Planning and Managing Water Systems, IFAC Workshop, Venice, Italy, 29 September 2004. [Google Scholar]
- Prasad, T.D.; Park, N.-S.; Asce, M. Multiobjective Genetic Algorithms for Design of Water Distribution Networks. J. Water Resour. Plan. Manag.
**2004**, 130, 73–82. [Google Scholar] [CrossRef] - Wang, Q.; Guidolin, M.; Savic, D.; Kapelan, Z. Two-Objective Design of Benchmark Problems of a Water Distribution System via MOEAs: Towards the Best-Known Approximation of the True Pareto Front. J. Water Resour. Plan. Manag.
**2015**, 141. [Google Scholar] [CrossRef][Green Version] - Kang, D.; Lansey, K. Revisiting Optimal Water-Distribution System Design: Issues and a Heuristic Hierarchical Approach. J. Water Resour. Plan. Manag.
**2012**, 138, 208–217. [Google Scholar] [CrossRef] - Liu, H.; Shoemaker, C.A.; Jiang, Y.; Fu, G.; Zhang, C. Preconditioning Water Distribution Network Optimization with Head Loss–Based Design Method. J. Water Resour. Plan. Manag.
**2020**, 146. [Google Scholar] [CrossRef] - Saldarriaga, J.; Páez, D.; León, N.; López, L.L.; Cuero, P. Power Use Methods for Optimal Design of WDS: History and Their Use as Post-Optimization Warm Starts. J. Hydroinform.
**2015**, 17, 404–421. [Google Scholar] [CrossRef][Green Version] - Prim, R. Shortest Connection Networks and Some Generalizations. Bell Labs. Tech. J.
**1957**, 36, 1389–1401. [Google Scholar] [CrossRef] - Ochoa, S. Diseño Optimizado de Redes de Distribución de Agua Potable Con Base En El Concepto Energético de Superficie Óptima de Gradiente Hidráulico (Ptimal Water Distribution Network Design Based on the Optimal Hydraulic Gradient Surface Energetic Concept); Universidad de los Andes: Bogotá, Colombia, 2009. [Google Scholar]
- Yusoff, Y.; Ngadiman, M.S.; Zain, A.-M. Overview of NSGA-II for Optimizing Machining Process Parameters. Procedia Eng.
**2011**, 15, 3978–3983. [Google Scholar] [CrossRef][Green Version] - Saldarriaga, J.; Bohorquez, J.; Celeita, D.; Vega, L.; Paez, D.; Savic, D.; Dandy, G.; Filion, Y.; Grayman, W.; Kapelan, Z. Battle of the Water Networks District Metered Areas. J. Water Resour. Plan. Manag.
**2019**, 145, 04019002. [Google Scholar] [CrossRef] - Wang, Q.; Wang, L.; Huang, W.; Wang, Z.; Liu, S.; Savić, D.A. Parameterization of NSGA-II for the Optimal Design of Water Distribution Systems. Water
**2019**, 11, 971. [Google Scholar] [CrossRef][Green Version] - Raad, D.N.; Sinske, A.N.; van Vuuren, J.H. Comparison of Four Reliability Surrogate Measures for Water Distribution Systems Design. Water Resour. Res.
**2010**, 46, W05524. [Google Scholar] [CrossRef] - Creaco, E.; Fortunato, A.; Franchini, M.; Mazzola, M.R. Comparison between Entropy and Resilience as Indirect Measures of Reliability in the Framework of Water Distribution Network Design. Procedia Eng.
**2014**, 70, 379–388. [Google Scholar] [CrossRef][Green Version] - Zhan, X.; Meng, F.; Liu, S.; Fu, G. Comparing Performance Indicators for Assessing and Building Resilient Water Distribution Systems. J. Water Resour. Plan. Manag.
**2020**, 146. [Google Scholar] [CrossRef] - Eliades, D.G.; Kyriakou, M.; Vrachimis, S.; Polycarpou, M.M. EPANET-MATLAB Toolkit: An Open-Source Software for Interfacing EPANET with MATLAB. In Proceedings of the 14th International Conference on Computing and Control for the Water Industry (CCWI), Amsterdam, the Netherlands, 7 November 2016. [Google Scholar]
- Jaramillo, A. Análisis de La Geometría Fractal de La Superficie Óptima de Presiones En El Diseño Optimizado de Redes de Distribución de Agua Potable; Universidad de los Andes: Bogotá, Colombia, 2020. [Google Scholar]
- Heinz-Otto, P.; Hartmut, J.; Dietmar, S. Chaos and Fractals, New Frontiers of Science; Springler: Berlin/Heidelberg, Germany, 2004; ISBN 978-1-4684-9396-2. [Google Scholar]
- Diao, K.D.; Butler, B.U. Fractality in Water Distribution Networks. In Proceedings of the 15th International Computing and Control for the Water Industry (CCWI), Sheffield, UK, 7 September 2017. [Google Scholar]
- Hwang, H.; Lansey, K. Water Distribution System Classification Using System Characteristics and Graph-Theory Metrics. J. Water Resour. Plan. Manag.
**2017**, 143, 04017071. [Google Scholar] [CrossRef] - Fujiwara, O.; Kang, D.B. A Two-Phase Decomposition Method for Optimal Design of Looped Water Distribution Networks. Water Resour. Res.
**1990**, 26, 539–549. [Google Scholar] [CrossRef] - Bragalli, C.; D’Ambrosio, C.; Lee, J.; Lodi, A.; Toth, P. On the Optimal Design of Water Distribution Networks: A Practical MINLP Approach. Optim. Eng.
**2012**, 13, 2019–2246. [Google Scholar] [CrossRef] - Reca, J.; Martínez, J. Genetic Algorithms for the Design of Looped Irrigation Water Distribution Networks. Water Resour. Res.
**2006**, 42, W05416. [Google Scholar] [CrossRef] - Yates, D.F.; Templeman, A.B.; Boffey, T.B. The Computational Complexity of the Problem of Determining Least Capital Cost Designs for Water Supply Networks. Eng. Optim.
**1984**, 7, 143–155. [Google Scholar] [CrossRef] - Artina, S. The Use of Mathematical Programming Techniques in Designing Hydraulic Networks. Meccanica
**1973**, 8, 158–167. [Google Scholar] [CrossRef]

**Figure 1.**Water distribution network classification flowchart. $\overline{D}$ represents the length-weighted average pipe diameter, $BI$ is the Branch Index, and $M{C}_{O-R}$ is the Meshedness Coefficient for the reduced network. Source: Hwang and Lansey [34].

**Figure 2.**Selected individuals from retrofitted OPUS/NSGA-II PFs for the: (

**a**) Hanoi WDS; (

**b**) Fossolo WDS; (

**c**) Balerma WDS; and (

**d**) Modena WDS.

**Figure 3.**Hanoi WDS with the corresponding HGL surfaces for the: (

**a**) [min(C), min(NRI)]; (

**b**) [knee(C), knee(NRI)]; and (

**c**) [max(C), max(NRI)] configurations.

**Figure 4.**Fossolo WDSs with the corresponding HGL surfaces for the: (

**a**) [min(C), min(NRI)]; (

**b**) [knee(C), knee(NRI)]; and (

**c**) [max(C), max(NRI)] configurations.

**Figure 5.**Balerma WDS with the corresponding HGL surfaces for the: (

**a**) [min(C), min(NRI)]; (

**b**) [knee(C), knee(NRI)]; and (

**c**) [max(C), max(NRI)] configurations.

**Figure 6.**Modena WDS with the corresponding HGL surfaces for the: (

**a**) [min(C), min(NRI)]; (

**b**) [knee(C), knee(NRI)]; and (

**c**) [max(C), max(NRI)] configurations.

**Figure 7.**WDSs’ fractal analysis criteria (${w}_{j}$) applied to a retrofitted OPUS/NSGA-II Hanoi PF.

**Figure 8.**WDSs’ fractal analysis criteria (${w}_{j}$) applied to a retrofitted OPUS/NSGA-II Balerma PF.

**Table 1.**Values of each parameter for the implementation of the NSGA-II and retrofitted approach. Source: Paez et al. [5].

Network | Individuals | Generations | Mutation Distribution Index | Crossover Distribution Index | Retrofitted Frequency |
---|---|---|---|---|---|

Hanoi | 500 | 500 | 20 | 3 | 5 |

Fossolo | 500 | 500 | 100 | 10 | 20 |

Balerma | 2000 | 4500 | 100 | 2 | 10 |

Modena | 2000 | 4000 | 20 | 7 | 50 |

Network | Reservoirs | Size | Pipes | Pipe Diameter Options | Search Space | Pressure Constraint | Velocity Constraint |
---|---|---|---|---|---|---|---|

Hanoi | 1 | Medium | 34 | 6 | $2.8\times {10}^{6}$ | ${\mathrm{P}}_{\mathrm{min}}=30\mathrm{m}$ | No |

Fossolo | 1 | Intermediate | 58 | 22 | $7.25\times {10}^{77}$ | P_{min} $=$ 40 m | Yes |

Balerma | 4 | Large | 454 | 10 | $1.00\times {10}^{455}$ | P_{min} $=$ 20 m | No |

Modena | 4 | Large | 317 | 13 | $1.32\times {10}^{353}$ | P_{min} $=$ 20 m | Yes |

**Table 3.**Cost and NRI for the three selected points of each network: [min(C), min(NRI)], [knee(C), knee(NRI)], and [max(C), max(NRI)].

Network | Point | Cost ($) | NRI(-) |
---|---|---|---|

Hanoi | min(C), min(NRI) | 6,439,320.50 | 0.222 |

knee(C), knee(NRI) | 7,260,699.00 | 0.304 | |

max(C), max(NRI) | 10,969,798.00 | 0.354 | |

Balerma | min(C), min(NRI) | 2,288,460.00 | 0.431 |

knee(C), knee(NRI) | 2,807,625.00 | 0.731 | |

max(C), max(NRI) | 13,191,652.00 | 0.891 | |

Fossolo | min(C), min(NRI) | 23,046.98 | 0.295 |

knee(C), knee(NRI) | 43,330.23 | 0.715 | |

max(C), max(NRI) | 1,661,922.50 | 0.999 | |

Modena | min(C), min(NRI) | 2,613,550.00 | 0.361 |

knee(C), knee(NRI) | 3,089,496.75 | 0.569 | |

max(C), max(NRI) | 6,731,936.00 | 0.907 |

**Table 4.**WDS classification, according to Hwang and Lansey [36]. $\overline{D}$ represents the length-weighted average pipe diameter, $BI$ is the Branch Index, and $MC$ is the Meshedness Coefficient for the reduced network.

Network | Point | Class | $\overline{\mathit{D}}$ (mm) | BI (−) | MC (−) |
---|---|---|---|---|---|

Hanoi | min(C), min(NRI) | Transmission Dense-Loop (TDL) | 682.952 | 0.438 | 0.333 |

knee(C), knee(NRI) | Transmission Dense-Loop (TDL) | 750.144 | |||

max(C), max(NRI) | Transmission Dense-Loop (TDL) | 1016.000 | |||

Balerma | min(C), min(NRI) | Distribution Branch (DB) | 173.259 | 0.770 | 0.052 |

knee(C), knee(NRI) | Distribution Branch (DB) | 189.439 | |||

max(C), max(NRI) | Transmission Branch (TB) | 446.902 | |||

Fossolo | min(C), min(NRI) | Distribution Dense-Grid (DDG) | 37.542 | 0.017 | 0.328 |

knee(C), knee(NRI) | Distribution Dense-Grid (DDG) | 57.947 | |||

max(C), max(NRI) | Transmission Dense-Loop (TDL) | 409.200 | |||

Modena | min(C), min(NRI) | Distribution Dense-Grid (DDG) | 145.779 | 0.033 | 0.331 |

knee(C), knee(NRI) | Distribution Dense-Grid (DDG) | 150.102 | |||

max(C), max(NRI) | Distribution Dense-Grid (DDG) | 264.884 |

Classification | Network | ${{\mathit{w}}_{\mathit{j}}}_{}$ | min(C), min(NRI) | knee(C), knee(NRI) | max(C), max(NRI) |
---|---|---|---|---|---|

Transmission Dense-Loop (TDL) | Hanoi | $\sum {d}_{ij}$ | 1.929 | 1.934 | 1.891 |

$HG{L}_{j}$ | 1.829 | 1.826 | 1.946 | ||

$\left(\sum {Q}_{ij}\right)-{D}_{j}$ | 1.020 | 1.020 | 1.020 | ||

Distribution Branch (DB), Transmission Branch (TB) | Balerma | $\sum {d}_{ij}$ | 1.857 | 1.807 | 1.798 |

$HG{L}_{j}$ | 1.798 | 1.798 | 1.791 | ||

$\left(\sum {Q}_{ij}\right)-{D}_{j}$ | 1.798 | 1.798 | 1.798 | ||

Distribution Dense-Grid (DDG) | Fossolo | $\sum {d}_{ij}$ | 1.950 | 1.829 | 1.829 |

$HG{L}_{j}$ | 2.033 | 2.047 | 0.874 | ||

$\left(\sum {Q}_{ij}\right)-{D}_{j}$ | 1.831 | 1.835 | 1.839 | ||

Modena | $\sum {d}_{ij}$ | 1.936 | 1.941 | 1.875 | |

$HG{L}_{j}$ | 2.040 | 2.044 | 2.082 | ||

$\left(\sum {Q}_{ij}\right)-{D}_{j}$ | 1.976 | 1.966 | 1.940 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Saldarriaga, J.; Salcedo, C.; González, M.A.; Ortiz, C.; Wiesner, F.; Gómez, S. On the Evolution of the Optimal Design of WDS: Shifting towards the Use of a Fractal Criterion. *Water* **2022**, *14*, 3795.
https://doi.org/10.3390/w14233795

**AMA Style**

Saldarriaga J, Salcedo C, González MA, Ortiz C, Wiesner F, Gómez S. On the Evolution of the Optimal Design of WDS: Shifting towards the Use of a Fractal Criterion. *Water*. 2022; 14(23):3795.
https://doi.org/10.3390/w14233795

**Chicago/Turabian Style**

Saldarriaga, Juan, Camilo Salcedo, María Alejandra González, Catalina Ortiz, Federico Wiesner, and Santiago Gómez. 2022. "On the Evolution of the Optimal Design of WDS: Shifting towards the Use of a Fractal Criterion" *Water* 14, no. 23: 3795.
https://doi.org/10.3390/w14233795